Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant

Kawohl, Bernhard ; Fridman, V.

Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 659-667 / Harvested from Czech Digital Mathematics Library

Résumé

First we recall a Faber-Krahn type inequality and an estimate for $\lambda_p(\Omega)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue $\lambda_p(\Omega)$ converges to the Cheeger constant $h(\Omega)$ as $p\to 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega$ which minimizes the ratio $|\partial D|/|D|$ among all simply connected $D\subset\subset\Omega$. As a byproduct we prove that for convex $\Omega$ the Cheeger set $\omega$ is also convex.

Détail

Publié le : 2003-01-01

MR 2062882 | Zbl 1105.35029

Classification:  35J20,  35J70,  49Q20,  49R05,  49R50,  52A38

Bibliographie

Alvino A.; Ferone V.; Trombetti G. On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal. 29 (1998), 437-451. (1998) | MR 1616519 | Zbl 0908.35094

Bhattacharia T. A proof of the Faber-Krahn inequality for the first eigenvalue of the $p$-Laplacian, Ann. Mat. Pura Appl. Ser. 4 177 (1999), 225-240. (1999) | MR 1747632

Belloni M.; Kawohl B. A direct uniqueness proof for equations involving the $p$-Laplace operator, Manuscripta Math. 109 (2002), 229-231. (2002) | MR 1935031 | Zbl 1100.35032

Belloni M.; Kawohl B. The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to \infty $, ESAIM COCV, to appear. | MR 2084254 | Zbl 1092.35074

Chavel I. Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives, Cambridge University Press, Cambridge, 2001. | MR 1849187 | Zbl 0988.51019

Cheeger J. A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis, A Symposium in Honor of Salomon Bochner, R.C. Gunning, Ed., Princeton Univ. Press, 1970, pp.195-199. | MR 0402831 | Zbl 0212.44903

Cicaclese M.; Trombetti C. Asymptotic behaviour of solutions to $p$-Laplacian equation, preprint No. 13, Univ. Napoli, 2002.

Demengel F. Theorémès d'existence pour des équations avec l'opérateur $1$-Laplacien, première valeur propre pour $-\Delta_1$, C.R. Acad. Sci. Paris, Ser. I 334 (2002), 1071-1076. (2002) | MR 1911649 | Zbl 1142.35408

Fridman V. doctoral thesis, in preparation, .

Giusti E. Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. | MR 0775682 | Zbl 0825.49059

Gonzales E.; Massari U.; Tamanini I. On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), 25-37. (1983) | MR 0684753

Huang Y.X. On the eigenvalues of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc. 125 (1997), 3347-3354. (1997) | MR 1403133 | Zbl 0882.35087

Juutinen P.; Lindqvist P.; Manfredi J. The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), 89-105. (1999) | MR 1716563 | Zbl 0947.35104

Kawohl B. Rearrangements and Convexity of Level Sets in PDE, Springer Lecture Notes in Math. 1150, 1985. | MR 0810619 | Zbl 0593.35002

Kawohl B. On a family of torsional creep problems, J. reine angew. Math. 410 (1990), 1-22. (1990) | MR 1068797 | Zbl 0701.35015

Kawohl B.; Stará J.; Wittum G. Analysis and numerical studies of a problem of shape design, Arch. Ration. Mech. Anal. 114 (1991), 349-363. (1991) | MR 1100800

Kawohl B.; Kutev N. Global behavior of solutions to a parabolic mean curvature equation, Differ. Integral Equations 8 (1995), 1923-1946. (1995) | MR 1348958

Kawohl B. Some nonconvex shape optimization problems, in: Optimal Shape Design, B. Kawohl et al., Eds., Springer Lecture Notes in Math. 1740 (2000), 7-46. | MR 1804684 | Zbl 0982.49024

Lindqvist P. On non-linear Rayleigh quotients, Potential Anal. 2 (1993), 199-218. (1993)

Lindqvist P. A note on the nonlinear Rayleigh quotient, in: Analysis, Algebra and Computers in Mathematical Research (Lulea 1992), M. Gyllenberg & L.E. Persson, Eds., Marcel Dekker Lecture Notes in Pure and Appl. Math. 156, 1994, pp. 223-231. | MR 1280948 | Zbl 0805.35085

Lindqvist P. On a nonlinear eigenvalue problem, Padova, 2000, pp. 79-110. | Zbl 0838.35094

Lefton L.; Wei D. Numerical approximation of the first eigenpair of the $p$-Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optim. 18 (1997), 389-399. (1997) | MR 1448898 | Zbl 0884.65103

Matei A.M. First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (2000), 1051-1061. (2000) | MR 1735181 | Zbl 0948.35090

Marcellini P.; Miller K. Elliptic versus parabolic regularization for the equation of prescribed mean curvature, J. Differential Equations 137 (1997), 1-53. (1997) | MR 1451535 | Zbl 0890.35046

Payne L.E.; Rayner M.E. An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z. Angew. Math. Phys. 23 (1972), 13-15. (1972) | MR 0313649 | Zbl 0241.73080

Payne L.E.; Rayner M.E. Some isoperimetric norm bound for solutions of the Helmholtz equation, Z. Angew. Math. Phys. 24 (1973), 105-110. (1973) | MR 0324202

Sakaguchi S. Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sci. Norm. Sup. Pisa (IV) 14 (1987), 404-421. (1987) | MR 0951227 | Zbl 0665.35025

Shirakawa K. Asymptotic convergence of $p$-Laplace equations with constraints as $p$ tends to $1$, Math. Methods Appl. Sci. 25 (2002), 771-793. (2002) | MR 1906858

Stredulinsky E.; Ziemer W.P. Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Anal. 7 (1997), 653-677. (1997) | MR 1669207 | Zbl 0940.49025